The Divisor Function for Matrices

We introduce a matrix divisor function τ_n(T,M), counting factorisations AB=M for n× n integer matrices A,B of height at most T. For a fixed non-singular M, or for the zero matrix M=O_n, we prove an asymptotic formula for τ_n(T,M), as T→∞, using lattice point counting. We also prove an essentially sharp uniform upper bound for τ_n(T,M), for an arbitrary non-singular matrix M.